Gamma trading tool

ABSTRACT

Automated hedging of financial instruments can include the automated generation of orders to hedge a financial exposure associated with a first financial instrument (e.g., to hedge a risk associated with the value of an option). The hedging orders include buy and sell orders to acquire long or short positions in a hedging instrument having a price movement that is correlated with price movements of the first financial instrument. The long and short positions are acquired so as to offset modeled changes in value of the first financial instrument. After an initial hedging order to buy or sell stock is filled by an exchange, subsequent hedging orders may be generated. Pricing and quantity for the subsequent hedging orders may be based on a user-specified movement in the price of the second financial instrument and may be automatically modified in response to price trending of the market with respect to the second financial instrument.

BACKGROUND OF THE INVENTION

[0001] An equity options trading desk of a financial institutionfrequently buys and sells options for customers and for proprietarytrading. An option C is a contract between two parties that gives thebuyer the right, but not the obligation, to purchase or sell a specifiedasset S (called the underlying asset) at a later date at an agreed uponprice K (called the strike price). An option to buy the asset isreferred to as a call, an option to sell the asset is referred to as aput. The purchase and sale of options involves several types of risk forthe investor. One of the risks associated with options is directionalrisk. Directional risk is a risk that the value of an option or aportfolio of options will be negatively affected by changes in the valueof the underlying asset. Financial institutions may reduce directionalrisk through a number of techniques, one of which is delta-hedging ofthe option position. Delta hedging involves buying and selling theunderlying asset based on the rate of change in the price of an optionwith respect to the price of the underlying asset (i.e., based on thedelta (Δ) of the instrument). Delta is defined as the first partialderivative of the option price C with respect to the stock price S. Inmathematical terms, delta is expressed as follows:${\Delta = \frac{\partial C}{\partial S}},$

[0002] Assuming that a delta value is constant, if the price of aparticular stock changes by an amount (X), then an option on that stockwill change by (Δ) times (X). Thus, if the delta of a call option is0.4, and the stock price changes by $1 then the option price is expectedto change by about 40% of that amount ($0.40). In this case, if aninvestor holds one call option and sells short A of the underlying stock(a “hedge position” on the investment), the investor will be immunizedagainst changes in the option price due to small movements of the stockprice.

[0003] In practice, the delta of an option is not constant. Thus, as thestock price changes, the delta likewise changes. When the change in thestock price becomes fairly large, the delta itself will changesignificantly. To maintain immunization against changes in the optionprice, the investor will need to adjust the hedge position. If, forexample, the trader established an initial hedge position at an initialdelta value of Δ₀ and the changed delta value becomes Δ₁, then thetrader will need to adjust the hedge position by the difference (Δ₀−Δ₁)between the original and new delta values. If the difference ispositive, the investor will need to purchase (Δ₀−Δ₁) shares of stock foreach option position; if the difference is negative, the investor willneed to sell (Δ₀−Δ₁) shares. This whole process of frequent adjustmentsis called delta-hedging and continues until option maturity or until thetrader sells the option back to the market.

[0004] As a practical matter, to apply delta hedging, a trader must beable to approximate the change in delta as the price of a stock changes.The change in delta may be approximated based on the Gamma (γ) of theoption. Gamma is a measure of the rate of change of delta with respectto changes in the price of the underlying asset. Gamma is defined as thefirst partial derivative of delta with respect to the stock price S. Byextension, γ is equivalently defined as the second partial derivative ofthe option price C with respect to the stock price S. In mathematicalterms, gamma is expressed as follows:${\Gamma = {\frac{\partial\Delta}{\partial S} = \frac{\partial C^{2}}{\partial^{2}S}}},$

[0005] For a change in stock price of (S₀−S₁), where S₀ is the initialstock price and S₁ is the new stock price, and assuming that the gamma(γ) of an option tends to remain relatively constant with respect tochanges in stock price during a trading day, the change in value of thedelta (which determines the quantity of stock to be purchased or sold tomaintain a hedge position) can be approximated as follow:

Δ₀−Δ₁≅γ×(S ₀−S₁)

[0006] If there is an unusually large change in stock price, or if theoption is close to maturity with S close to K, the assumption of aconstant gamma may be inappropriate and intra-day adjustments to thegamma value may be required.

[0007] If a trader has bought a plain call or put option, then the gammaof the option position is positive (commonly referred to as “longgamma”). FIG. 1 shows a simple long gamma position for a plain vanillaEuropean Call option with a strike price K=100, a time to maturity T=1year, on a stock with a current price S=100, a volatility σ=10%, and ariskless rate r=5% (riskless rates are the government bonds interestrates). Line 101 (“C”) shows the price of the call option with respectto stock price S (shown as the X axis). The price of the option 101forms a curve and a measure of the curvature (i.e. non-linearity) of theline at any particular point is its gamma. Line 104 shows the value ofan initial quantity of stock (Δ) that was sold short by the trader tohedge the option position. The value of the quantity of stock (Δ) soldshort 104 is shown as decreasing linearly when the option price isincreasing, and increasing linearly when the option price is decreasing.This linear behavior of (Δ) provides a relatively good hedge for a smallmovement of the stock price.

[0008] Line 103 shows the combined value of the long option position 101and the short A stock 104 when there is no rebalancing of the shortposition Δ 104. Due to the curvature of the option price given by thegamma, this combination will produce a profit (line 102). This profit102 is usually called long gamma trading profit. If, when the optionmatures, this profit is greater than the premium paid to buy the calloption, then the trader will have made a gain, otherwise there will be aloss. A gamma trading strategy tries to maximize the profit or minimizethe loss generated by the delta hedging of an option position.

[0009] For a gamma trading strategy, the magnitude and frequency of theswings of the stock price (which can be measured in relation to aninitial reference price such as the initial stock price “S” or theinitial strike price “K”) typically has a more significant effect onprofit and loss than does the direction of the stock movement. Thismagnitude and frequency is measured by the realized volatility of thestock during the lifetime of the option, (σ_(r)). If the realizedvolatility (σ_(r)) is greater than the volatility implied by the optionprice when the option was bought (σ), then the strategy will have aprofitable outcome. It is noted that implied volatility (σ) may becalculated using the well-known Black-Scholes model.

[0010] If a trader is primarily interested in closely hedging severaloption positions each with a long gamma, the trader has to frequentlymonitor the price of the several underlying stocks and rebalance thehedge position for each stock by buying or selling an amount of stock(Δ₀−Δ₁) in response to changes in each stock price. Manual re-balancingof a hedge position in this manner can be time consuming as it requiresthe option trader to closely monitor the prices of all the stocks in thetrader's portfolio. Furthermore, to ensure that desired hedging tradesare executed, a trader may resort to buying at the best offer price(rather than placing a limit order at the lower best bid price), orselling at the best bid price (rather than placing a limit order at ahigher best offer price). This buying at the best offer price andselling at the best bid price, referred to as paying the bid-ask spread,may reduce the trader's profitability.

[0011] Delta hedging and gamma trading are explained in greater detailin, e.g., Baz Jamil, Vasant Naik, David Prieul, Vlad Putyatin, FrancisYared, Selling risk at a premium, Risk, December 2000, pages 135-138;Hull John, Introduction to Futures and Options Markets, second edition,Prentice Hall International Edition, 1995, pages 319-345.

SUMMARY OF THE INVENTION

[0012] Automated hedging of financial instruments is provided bycomputer-implemented systems and methods. The systems and methodsprovide for the automated generation of hedging orders to hedge afinancial exposure associated with a first financial instrument (e.g.,to hedge a risk associated with the value of an option). The hedgingorders include buy and sell orders to acquire long or short positions inanother financial instrument having a price movement that is correlatedwith price movements of the first financial instrument. The long andshort positions are acquired to offset changes in value of the firstfinancial instrument. After an initial buy or sell hedging orders isfilled by an exchange, subsequent hedging orders may be generated.Pricing and quantity for the subsequent hedging orders may be based on auser-specified movement in the price of the second financial instrumentand may be automatically modified in response to price trending of themarket with respect to the second financial instrument.

[0013] Implementations may provide for continuous monitoring of hedgingorders, automatic slicing of large orders into smaller orders (e.g., toavoid showing a big position to the market), customization of hedgingstrategy on a per-stock/per-account basis, assisting an option trader inperforming effective market-making for a stock, avoidance of having topay bid-ask spread, and trading based on intraday stock market-makingactivity to increase trading profit. The details of one or moreembodiments of the invention are set forth in the accompanying drawingsand the description below. Other features, objects, and advantages ofthe invention will be apparent from the description and drawings, andfrom the claims.

DESCRIPTION OF THE DRAWINGS

[0014]FIG. 1 is a graph of stock and option pricing data.

[0015]FIG. 2 is a trading system diagram.

[0016]FIG. 3 is a trading interface.

DETAILED DESCRIPTION OF THE INVENTION

[0017]FIG. 2 shows a trading platform 200 that provides real-timetrading and order management features. The system 200 includes one ormore trader terminals 201 providing an interface with which traders canview trading data and place trades to be executed by destinationexchanges. Traders can also enter data at the terminals 201 to initiatethe automatic generation of hedging orders by the platform 200.

[0018] The trading terminals 201 may be personal computers with anetwork interface enabling communications with other system components202-205 (e.g., computers executing the Microsoft Windows operatingsystem) and that are software-configured to provide an interface withwhich trading data can be displayed to a trader and trades entered.Communication between the terminal 201 and systems 202-205 may be bymeans of industry standard and/or proprietary protocols. For example,trading information may be communicated between the terminal 201 andother systems 202-205 using the Financial Information Exchange (FIX)message protocol. The FIX message protocol specifies command messagesthat are understood by stock exchange order entry systems. A detailedFIX protocol specification is available from the FIX ProtocolOrganization web site at www.fixprotocol.org. Alternatively, aproprietary protocol can be used by terminal 201 and subsequentlyreformatted or translated using a message protocol understood by adestination exchange. In the example implementation that follows, theterminal 201 uses a proprietary protocol whereby a simple text stringcontaining trading data is generated and sent to a gateway computer 202(e.g., “Buy 100 shares IBM at 100”). At the gateway computer 202,additional parameters may be added and further message formatting may beperformed. For example, the trade order data from the terminal 201 maybe formatted in accordance with the FIX protocol at the gateway 202.

[0019] After a FIX message is generated at the gateway 202, it is sentto an order management and routing system (OM-OR) 203. The OM-OR systemmaintains both the current and previous state of an order. For example,the OM-OR can submit orders over a network 206 to one or moredestination exchanges, track the portion of an order that has beenexecuted and, if a later instruction is received to reduce order size,the OM-OR will use the data about the executed portion to determine theamount by which the outstanding order portion is to be reduced. TheOM-OR also may slice large orders into sizes acceptable by theparticular exchange, add additional slice(s) to the order if an ordersize is increased, transmit order modification requests to destinationexchanges, manage modification requests to ensure that they followexchange rules, enable modification and cancellation of orders not yetreleased to an exchange, support timed release of orders, and translateor format FIX protocol message into an exchange link protocolscompatible with a particular destination exchange (e.g., the GL API,Exchange API, or Fidessa protocols).

[0020] The OM-OR also receives real-time security pricing informationfrom one or more exchanges. For securities listed on multiple exchanges,this information can be used to choose the exchange with the best pricefor a particular order. Such real-time information, as well as otheraccount, order, and status data is accessible by the terminal 201through messages exchanged between the terminal 201 and gateway 202 ordirectly between terminal 201 and the OM-OR 202. For example, theterminal 201 can query the OM-OR to determine all orders and theircurrent execution state for a given account, exchange or range ofexecution identifiers.

[0021] To assist in the hedging of options portfolios, the OM-OR isconnected to a hedging engine 204. The hedging engine 204 canautomatically generate and place limit buy and sell orders withexchanges at calculated hedging levels. The hedging strategy implementedby the hedging engine 204 can be configured by a trader using a numberof adjustable parameters. Thus, through the use of these parameters, theway in which the hedging engine reacts to market price volatility can bemodified. The hedging strategy parameters, as well as other tradingdata, may be viewed and entered by a trader using the terminal 201. Inone implementation, the terminal provides interface 300 (FIG. 3) for thedisplay and entry of data. Other interfaces may also be used, e.g., theinterface can be command-line based or a graphical user interface otherthan the one shown in FIG. 3. The trading terminal interface can displaydata about a trader's portfolios and the stocks used to hedge hisdirectional risk, and can receive inputs from the trader to createcustom hedging strategies for particular stock/portfolio combinations.For example, for each stock in each account managed by a trader, thetrader may specify a different percentage price move that will trigger ahedging trade. Thus, in order to capture as much stock volatility aspossible and maximize the profit, a small percentage move may bespecified if the stock price is mean-reverting, and a large percentagemove if the stock is trending. The trader can also introduce asymmetryaround the reference price so that, after a few consecutive executionson the same side (buy or sell), the percentage move for the subsequenthedge is increased on the executed side and decreased on the other. Thetrader also can decide to hedge only a percentage of the delta change,thus taking a directional view.

[0022] In the FIG. 3 implementation, each row 301-308 of the tradingterminal interface contains data specifying a hedging strategy for adifferent underlying stock/account combination. For example, row 301specifies a hedging strategy that applies to the underlying stockidentified by symbol “aah.as” in the account “7400J6E.” Since eachhedging strategy (i.e., each row 301-308) is specific to both anunderlying stock and an account, a trader can trade the same instrumentin different accounts using different strategies. To establish a hedgingstrategy, the trader must first specify particular underlying stocks inparticular accounts he wants to use to hedge. In some implementations,this may be done by querying a risk management and position keepingapplication that will return the option's risk and volatility statistics(i.e., values of Δ, γ, and σ) aggregated by underlying stock (column 330“Stock”) and account (column 331 “Account”).

[0023] The currency Δ (Δ multiplied by the price of the stock in localcurrency) of the option held on the underlying stock is shown in column333 as the “Exposure.” For example, the value “−51,306” in the Exposurecolumn 333 of the row 301 indicates that 51,306 EUR of the stock“aah.as” were sold short in the account “7400J6E”. The “Exposure” valueis a measure of the initial directional risk associated with aparticular stock/account combination. Other values that areautomatically displayed at the interface include a reference price 332(labeled “Close Price” in FIG. 2) and the “Gamma” 334. The gamma tradingengine 204 uses the reference price 332 as a point about which swings instock price are measured. The reference price may actually be set basedon one of a number of different prices (e.g., the open, the current midprice (i.e., the average of the current buy and current ask price),previous session close, or the last trade price). Using radio buttons310, the trader can specify, on a per-stock/per-account basis, whetherto use the open, current mid, close or last price as the reference price332.

[0024] The trading engine uses the “Gamma” value to determine the numberof shares that must be bought or sold to maintain effective hedging. TheGamma value shown at interface 200 is expressed as the local currency(i.e. the currency in which the stock price is expressed) change invalue of an option for a 1% move in the underlying stock price. Forexample, the Gamma value of “111,167” in row 301 means that for a 1%move in the price of the stock aah.as, the change in the value of anoption on the stock “aah.as” in account 7400J6E is EUR 111, 167.

[0025] Other values displayed to the trader, or input by the trader,using the interface 300 include Pct G, Volatility, Vol Offset, Buy Qty,Buy Price, Sell Price, Sell Qty, Resend, Skew, and Slice (shown incolumns 335-344, respectively).

[0026] The Pct G value (Percentage Gamma, column 335) specifies thepercentage of the Gamma that the trader wishes to hedge. The Volatilityvalue (column 336) specifies the Gamma weighted average of the annualimplied volatilities of every option held on the underlying stock. ThisGamma value may be automatically obtained from a risk management system(which may be part of the hedging engine 204). For example, thevolatility value of 28.6 in column 335 row 301 means that, in the nextyear, the price of the instrument “aah.as” may rise or fall within anestimated ±28.6% range of the Close Price. The Vol Offset value enablestrader-modification of the “Volatility” value by allowing manualincreases or decreases of the Volatility value. As further explained,below, the value of Volatility plus Vol Offset is used by the gammatrading engine 204 to automatically calculate a percMove(σ) parameter byconverting the annual implied volatility to a daily volatility.Different formulas may be selected to convert from annual to dailyvolatility using radio buttons 315 or 316 and the NonBusinessDayparameter 317.

[0027] If button 315 (“sqrt(252)”) is selected, then the dailyvolatility is calculated as:${{{percMove}(\sigma)} = {\frac{{Volatility} + {VolOffset}}{\sqrt{252}} \times \sqrt{NonBusinessDay}}};$

[0028] If button 316 (“sqrt(7/5)”) is selected, then the dailyvolatility is calculated as:${{{percMove}(\sigma)} = {\frac{{Volatility} + {VolOffset}}{\sqrt{365}} \times \sqrt{\frac{7}{5}} \times \sqrt{NonBusinessDay}}};$

[0029] In addition to trading at the percMove (σ) point, the gammatrading system permits a trader to manually set an additional percentagevalue at which trading is to occur. This additional trading percentagevalue is input as the Resend value 342. The Resend value 342 correspondsto the fixed percMove value used by the gamma trading algorithm, below.Thus, for example, if the reference stock price is $100, the dailyvolatility is 10% (percMove(σ)), and the Resend value (percMove) is 5%,then the gamma trading engine will generate a buy order at $90 and asell order at $110 (the 10% values). If, for example, the buy order isfully executed, the gamma trading engine will then cancel anyoutstanding sell orders, establish a new reference price of $90, and setnew buy and sell orders at 95% and 105% of the new reference price. TheSkew value 343 corresponds to the “asym” parameter explained in thegamma trading algorithm, below. The Slice value 344 is used to determinethe maximum size of a buy or sell order to be delivered to an exchange.Values of “Buy Qty”, “Buy Price” “Sell Price” and “Sell Qty” (columns338-341, respectively) are automatically calculated and displayed at theterminal 301. This automatic calculation may be performed at theterminal 201 using the following formulas:

Buy Price:=Reference Price*(100−percMove(σ))/100.

Sell Price:=Reference Price*(100+percMove(σ))/100.

Buy Qty:=(percMove(σ)*Gamma)/Buy Price.

Sell Qty:=(percMove(σ)*Gamma)/Sell Price.

[0030] In other implementations, these values may be calculatedelsewhere in the system 300 (e.g., hedging engine 204) and provided tothe terminal 201.

[0031] After the desired inputs are set, the trader can submit selectedones of the trading strategies to the hedging engine 204. This can bedone by selecting particular input rows 301-308 corresponding to thestock/account combinations that the trader wishes to hedge, and thenselecting the “Send Trades” button 320. The relevant strategy data isthen sent to the hedging engine 204 (either directly or, e.g., viagateway 202). The hedging engine 204 uses the received strategyparameters, as well as real-time stock pricing data received overnetwork 206 from stock exchanges, and order execution data managed bythe OM-OR 203, to generate and manage the creation of buy and sellorders in a stock. This generation and management is performed using agamma trading algorithm. The following is an implementation of the gammatrading algorithm, though other algorithms and modifications thereof mayalso be used (algorithm is shown in pseudo-code):LongGammaHedge(percΓ,refP,percMove(σ),percMove,maxSize, asym) { buyPrice:= refP * (100 - percMove(σ)) / 100; sellPrice : = refP * (100 +percMove(σ)) / 100; buyQty : = (percMove(σ) * percΓ) / buyPrice; sellQty:= (percMove(σ) * percΓ) / sellPrice; cumAsym := 0; do { qtyBought := 0;qtySold := 0; posDelta := 0; /* delta after one side is completelyexecuted (almost always 0) */ buy min(maxSize, buyQty - qtyBought) ofstock at buyPrice; sell min(maxSize, sellQty - qtySold) of stock atsellPrice; do { wait for full fill; if fill is on buy side { posDelta :=posDelta + fillQty; qtyBought := qtyBought + fillQty; cumAsym :=cumAsym - asym; buy min(maxSize, buyQty - qtyBought) of stock atbuyPrice; } else { posDelta :=posDelta - fillQty; qtySold := qtySold +fillQty; cumAsym := cumAsym + asym; sell min(maxSize, sell Qty -qtySold) of stock at sellPrice; } while qtyBought < buyQty andqtySold <sellQty; /* if the buy side was completely filled */ if (qtyBought =buyQty) /* if we received a partial fill on the sell side we will beshort posDelta shares */ posDelta := posDelta - buyQty; else /* if wereceived a partial fill on the buy side we will be long posDelta shares*/ posDelta :=posDelta + sellQty; cancel the unfilled order; refP :=fill price; asymToUse := 0; /* if we had at least two fills on a row onthe same side then asymToUse <> 0 */ if cumAsym> asym asymToUse :=cumAsym - asym; else if cumAsym < -asym asymToUse := cumAsym + asym;buyPrice := refP * (100 − percMove + asymToUse) / 100; sellPrice :=refP * (100 + percMove + asymToUse) / 100; buyQty := min[(percMove *percΓ) / buyPrice - posDelta, 1]; sellQty := min[(percMove * percΓ) /sellPrice + posDelta, 1]; } while trader doesn't cancel the orders andabs(asymToUse) < percMove; }

[0032] For each hedging strategy, (i.e., each submitted row 301-308) thefollowing starting inputs are provided to the hedging algorithm:

[0033] percγ: The percγ parameter corresponds to the “Gamma” value ofcolumn 334 multiplied by the “Pct G” value of column 335.

[0034] refP: The refP (reference price) parameter corresponds to the“Close Price” value of column 332. The refP value may be the close,opening, last trade or current mid price of the stock.

[0035] percMove (σ): The percMove (σ) parameter corresponds to the dailyvolatility, calculated from the Volatility value (column 336) and theannual to daily volatility conversion algorithm configured by input315-317. PercMove(σ) specifies the percentage movement in the underlyingstock price at which the trader wants to hedge. For an initial trade,percMove(σ) is a function of the implied volatility of the options onthe stock. For subsequent trades, this value may be fixed by the trader(th is trader-specified value represented in the gamma trading algorithmas “percMove”). The subsequent trade value, percMove, corresponds to the“Resend” value (column 342) entered by the trader.

[0036] maxSize: The maxSize parameter specifies the maximum size of atrade that a trader wants to show to the market. The maxSize iscalculated as a function of the initial buy or sell order sizedetermined by the gamma trading algorithm and the Slice value (column344). In some implementations, the trader may directly input the maxSizevalue (e.g. as an alternative to the number of slices value in column344).

[0037] asym: The asym parameter (column 343) specifies an asymmetry inthe spread around the reference price refP. The asym parameter can beused to modify the trading strategy in cases where the market price fora particular stock is trending.

[0038] The operation of this algorithm will now be explained by example.Consider, for example, the execution of the gamma trading algorithm withthe following parameter values: percγ=1000, refP=100, percMove(σ)=3%,percMove=1.5%, maxSize=16, asym=0.5). When the hedging algorithm isprocessed by the engine 204, an initial sequence of buy/sell hedgingorders is generated (shown as “1^(st) Hedge Sequence”), below. 1^(st)Hedge Sequence buy Qty=31, sell Qty = 29; buy 16 stock at 97;  // Submit1^(st) slice of the first buy order to the exchange. sell 16 stock at103;   // Submit 1^(st) slice of the first sell order to the exchange.filled on buy 16 stock at 97; // 1^(st) slice of the first buy orderfilled by the exchange. buy 15 stock at 97;  // Submit 2^(nd) slice ofthe first buy order to the exchange filled on buy 15 stock at 97; //2^(nd) slice of the first buy order filled by the exchange. cancel sellorder 16 of stock at 103; // Cancel outstanding sell order.

[0039] As shown by the “1^(st) Hedge Sequence”, based on the initialparameter data, the gamma trading algorithm will determine that for aparticular underlying stock the appropriate hedging buy quantity is 31shares at a price of 97, and the sell quantity is 29 shares at a priceof 103. However, the maxSize parameter limits the maximum trading ordersize to 16. Therefore, the buy and sell quantities will be submitted tothe exchange as a number of different segments (in this case, two).Assuming that the first buy order is filled (i.e., executed by anexchange), a second buy order is generated for the remaining fifteenshares and submitted to the exchange. If the second buy order is thenfilled, the outstanding sell order is cancelled.

[0040] After the first hedging sequence has been completed (eitherbecause the entire sell or buy order was executed), a new referenceprice is automatically determined. Where, as here, the hedge wascompleted through filling of the buy order, the next reference pricewill be the buy price at which the order was filled (i.e., 97).Similarly, if the hedge was completed through the filling of the sellorder, the next reference price will be the sell price at which theorder was filled, and the volatility percentage percMove (i.e., the“Resend” value of column 342) controls the generation of hedging orders.Using this data, and based on the completion of the 1st Hedge sequenceas described above, the following hedge sequence is generated: The2^(nd) Hedge Seciuence refP=97, buyQty= 16, sellQty = 15, asymToUse=0;buy 16 stock at 95.55; sell 15 stock at 98.45; filled on buy 16 stock at95.55; cancel sell order 15 of stock at 98.45;

[0041] For this 2^(nd) Hedge Sequence, the gamma trading algorithmdetermines that the buy quantity is to be set to 16 and the sellquantity to 15. Here, as with the 1^(st) Hedge Sequence, the orders arefilled on the buy side.

[0042] A third hedge sequence may then be generated. This third exampleillustrates the effect of a non-zero asymToUse value (i.e., a non-zeroSkew value in column 343). 3^(rd) Hedge Sequence refP=95.55, buyQty=21,sellQty = 10, asymToUse=−0.5%; buy 16 stock at 93.64;   //1^(st) slicesell 10 stock at 96.51; filled on buy 16 stock at 93.64; buy 5 stock at93.64;   //2^(nd) slice filled on buy 5 stock at 93.64; cancel sellorder 10 of stock at 96.51;

[0043] In the following hedging sequences the asymToUse value is furtherincreased based on the Skew (column 343) until the algorithm terminates:4^(th) Hedge Sequence+HZ,1/32 refP=93.64, buyQty=27, sellQty = 5,asymToUse=−1%; buy 16 stock at 91.3;   //1^(st) slice sell 5 stock at94.1; filled on sell 5 stock at 94.1; cancel buy order 16 of stock at91.3; 5^(th)Hedge Sequence refP=94.1, buyQty=22, sellQty = 11,asymToUse=− 0.5%; buy 16 stock at 92.22;   // 1^(st) slice sell 11 stockat 95.04; filled on buy 16 stock at 92.22; buy 6 stock at 92.22;   //2^(nd) slice filled on buy 6 stock at 92.22; cancel sell order 11 ofstock at 95.04; 6^(th) Hedge Sequence refP=92.22, buyQty=28, sellQty =5, asymToUse=1%; buy 16 stock at 89.91;   // 1^(st) slice sell 5 stockat 92.68; filled on buy 16 stock at 89.91; buy 12 stock at89.91;   //2^(nd) slice filled on buy 12 stock at 89.91; cancel sellorder 5 of stock at 92.68;

[0044] 7_(th) Hedge Sequence

[0045] asymToUse=−1.5% which is equal to percMove, therefore STOP Insome implementations, the gamma trading algorithm can include a numberof enhancements for “real market” functioning. For example, animplementation may include checks on the min and max quantity to buy andsell, rounding on the ticksizes and the lotsizes, a delta adjustment tohandle transactions in which the trading engine processes a full fill onone side and a partial fill on the other side. Furthermore, thealgorithm may be implemented in an event-driven manner to respond toevents generated by the trader or arising due to market and exchangeactivity. The following are some example events that may drive thealgorithm:

[0046] Events generated in response to activity by a trader include:

[0047] 1. New Order—A new trading order or hedging strategy is input bya trader.

[0048] 2. Cancel Request—The trader request cancellation of a previouslyentered trading order or hedging strategy.

[0049] 3. Replace an old Order with a new Order—The trader requestreplacement or modification of an outstanding trading order or hedgingstrategy.

[0050] Events generated in response to market and exchange activity caninclude:

[0051] 1. Order Acknowledged—Notification is received from an exchangeacknowledging the insertion of a trading order in the exchange's orderbook.

[0052] 2. Order Partially Filled—Notification is received from anexchange that an order has been partially filled.

[0053] 3. Order Filled—Notification is received from an exchange thatthe order has been completely filled.

[0054] 4. Order Done For Day—Notification is received from an exchangethat an order has been cancelled from the order book due to thetermination of the trading day.

[0055] 5. Order Cancelled—Notification is received from an exchange thatan order has been cancelled from the order book.

[0056] 6. Order Replaced—Notification is received from an exchange thatan old order has been replaced with a new one.

[0057] 7. Order Pending Cancel—Notification that a request to cancel anorder has been put on hold by an exchange.

[0058] 8. Order Stopped—Notification is received from an exchange thatthe trading on the order's stock has been stopped.

[0059] 9. Order Rejected—Notification is received from an exchange thatan order has been rejected.

[0060] 10. Order Suspended—Notification is received from an exchangethat the trading in the stock for which we have an order in the orderbook has been suspended.

[0061] 11. Order Pending New—Notification is received from an exchangethat the request to insert a new order in the order book has been put onhold.

[0062] 12. Order Expired—Notification is received from an exchange thatthe lifetime of the order has expired.

[0063] 13. Order Restated—Notification is received from an exchange thatone or more of the attributes of order (i.e. quantity, price, . . . )has changed.

[0064] 14. Order Pending Replace—Notification is received from anexchange that the replace request has been put on hold.

[0065] 15. Order Cancel Rejected—Notification is received from anexchange that a cancel request has been rejected is received from anexchange.

[0066] In response to the foregoing events, and/or due to other changesin market, account, and order status, the interface 300 can be updatedin real-time. For example, as trades are made, gateway 202 may transmitsdata to terminal 201 to cause the terminal 201 to display updatedinformation regarding executions occurring for each stock/accountcombination.

[0067] The invention may be implemented in digital electronic circuitry,or in computer hardware, firmware, software, or in combinations of them.Apparatus of the invention may be implemented in a computer programproduct tangibly embodied in a machine-readable storage device forexecution by a programmable processor; and method steps of the inventionmay be performed by a programmable processor executing a program ofinstructions to perform functions of the invention by operating on inputdata and generating output. The invention may advantageously beimplemented in one or more computer programs that are executable on aprogrammable system including at least one programmable processorcoupled to receive data and instructions from, and to transmit data andinstructions to, a data storage system, at least one input device, andat least one output device. Each computer program may be implemented ina high-level procedural or object-oriented programming language, or inassembly or machine language if desired; and in any case, the languagemay be a compiled or interpreted language. Suitable processors include,by way of example, both general and special purpose microprocessors.Generally, a processor will receive instructions and data from aread-only memory and/or a random access memory. Storage devices suitablefor tangibly embodying computer program instructions and data includeall forms of non-volatile memory, including by way of examplesemiconductor memory devices, such as EPROM, EEPROM, and flash memorydevices; magnetic disks such as internal hard disks and removable disks;magneto-optical disks; and CD-ROM disks. Any of the foregoing may besupplemented by, or incorporated in, specially-designed ASICs(application-specific integrated circuits).

[0068] A number of embodiments of the present invention have beendescribed. Nevertheless, it will be understood that variousmodifications may be made without departing from the spirit and scope ofthe invention. For example, automatic estimates of the hedge levels maybe based on time series data of the underlying stock and risk parameters(Δ, γ, σ) may be calculated and updated in real-time to allow moreprecise hedging. The invention may also be applied to hedge other typesof financial instruments having a value that is related to an underlyingstock or traded instrument. For example, convertible bonds may be hedgedusing the invention. In the case of convertible bonds, the risk systemand interface treat the convertible bond the same as a plain vanillaoption (i.e., risk is assessed based on the underlying stock and theunderlying stock is what is shown on the interface 300). Accordingly,other embodiments are within the scope of the following claims.

What is claimed is:
 1. A computer-implemented method of financialinstrument trading, the method comprising: calculating a financial riskassociated with a first financial instrument based on a pricingvolatility model; generating an order to hedge the first financialinstrument by acquiring a second financial instrument at a target pricethat is determined based on the pricing volatility model; andtransmitting the order to an exchange.
 2. The method of claim 1 whereinacquiring the second financial instrument comprises acquiring a positionselected from the group consisting of a short position and a longposition in the second financial instrument.
 3. The method of claim 1wherein the order to hedge comprises an order to acquire a secondfinancial instrument having a valuation behavior acting to offsetvaluation change of the first financial instrument.
 4. The method ofclaim 1 wherein: the order comprises a buy order at a target price lessthan the current asking price; the method further comprises generating asecond order to hedge comprising a sell order to sell the secondfinancial instrument at a target price greater than the current bidprice for the second financial instrument, and transmitting the orderfurther comprises transmitting the buy and the sell order to theexchange such that both the buy and the sell order are simultaneouslypending and execution of the buy or sell order is determined based onprice movement of the second financial instrument.
 5. The method ofclaim 1 further comprising automatically generating a subsequent orderto acquire the second financial instrument in response to a change inthe pricing volatility model.
 6. The method of claim 5 wherein the firstfinancial instrument comprises an option on an underlying stock and thesecond financial instrument comprises the underlying stock.
 7. Themethod of claim 6 wherein the model is based on a first derivative ofthe option price with respect to a first derivative of the underlyingstock price.
 8. The method of claim 1 wherein the financial riskcomprises a risk associated with pricing volatility of the firstfinancial instrument.
 9. A financial instrument trading systemcomprising: a terminal interface coupling the system to a plurality oftrading terminals; a trading network interface coupling the system to afinancial instrument trading exchange; a hedging engine operativelycoupled to the terminal interface and to the trading network interfaceand configured to receive hedging data from ones of the tradingterminals and to transmit buy and sell hedging orders to the financialinstrument trading exchange, the hedging engine comprising storedinstructions to configure generation of successive stages of buy andsell hedging orders, each of the buy and sell orders comprising a pricedetermined based on a volatility model associated with a financialinstrument and a reference price for the financial instrument, thereference price being adjustable for each of the successive stages. 10.The system of claim 9 wherein the system further comprises a portfoliomanagement system and the hedging engine further comprises storedinstructions to configure the hedging engine to retrieve dataidentifying each of a plurality of financial instruments held in a firstaccount and to assess risk associated with each of the plurality offinancial instruments based on the volatility model.
 11. Acomputer-implemented method of financial instrument trading, the methodcomprising: calculating a financial risk associated with a firstfinancial instrument based on a pricing volatility model; in a firsthedging stage, generating a first buy order and a first sell order eachfor a second financial instrument having a valuation that is correlatedwith the valuation of the first financial instrument, the first buyorder comprising a buy target price and a buy quantity and the firstsell order comprising a sell target price and a sell quantity, the buyand the sell target price and the buy and the sell quantity each beingautomatically determined based on an initial volatility value determinedfrom the pricing volatility model so as to enable hedging of thefinancial risk; transmitting the first buy and the first sell order toan exchange; receiving a notification from the exchange that one of thefirst buy order or the first sell order has been filled by the exchangeand then canceling the unfilled one of the first buy order or first sellorder; and in each of a plurality of subsequent hedging stages,generating another buy and another sell order each for the secondfinancial instrument based on a volatility value received from a traderand on a reference price at which a previous buy or previous sell orderwas filled by the exchange during a previous hedging stage and on amodel adjusting the buy or sell order based on trending of the marketwith respect to the second financial instrument; transmitting said buyand said sell order to the exchange; receiving a notification from theexchange that one of said buy or sell orders has been filled by theexchange and then canceling the unfilled one of the buy order or sellorders.
 12. The method of claim 11 wherein the first instrumentcomprises an option on an underlying stock and the second instrumentcomprises the underlying stock.